Drive signals, such as motor drive signals, can excite undesirable vibrational responses in the systems to which they are applied, such as servomechanisms and servomotors. Such vibrational responses are often the limiting factor in motion control designs, especially when attempting to obtain rapid actuation. Conventional methods for generating motion commands, from which the drive signals are generated, attempt to avoid exciting large vibrational response in mechanical systems in a number of ways. One common approach is to constrain the command formulation in the time-domain to produce a “smooth” command profile that limits the bandwidth of the acceleration spectrum. These methods often employ piecewise differentiable functions that have continuous derivatives up to a specified order. By limiting the magnitude of the derivatives, the bandwidth of the acceleration command is typically reduced.
Another common method of command generation relies on timing the motion duration to match the natural vibration period of the most sensitive mode. For commands where the initial and final velocities are sufficiently close, the acceleration signal will have odd symmetry about the midpoint of the command. The resulting motion command will then excite the vibration mode during the first half of the command and then remove vibrational energy during the second half of the command.
A third category of motion command creation relies on filtering to constrain the acceleration spectrum of a command signal. One particular variation of this approach employs a notch filter to remove frequency content from the command that would otherwise excite a system resonance and result in unwanted vibration.
While most of the traditional approaches have been shown to be beneficial at reducing unwanted vibration, there are limitations to each of these methods. None of these techniques strikes a good balance between minimizing the time required to execute the motion, controlling the spectral content of the commanded acceleration, and maintaining robustness to frequency variations in the response vibration mode. The approach of formulating smooth commands in the time domain does not tightly control the spectral content of the commanded acceleration. Such results in motion having excessive energy at critical frequencies or which takes too long to execute. The technique of timing the slew duration to match the dominant vibration period can be highly effective. But this method is not very robust to even small uncertainty in the vibration period or to shifts in the resonant frequency of vibrational modes. Moreover, this approach breaks down or becomes algorithmically much more complicated if the velocities at the beginning and end of the command are dissimilar to the point of losing predominately odd symmetry in the acceleration signal. The command filtering approach can effectively remove unwanted frequency content from the command, but the commanded duration will be lengthened by the transient response of the filter. This translates to a motion command that requires significantly more time to execute than originally intended.
In light of the shortcomings of these and other techniques, the need has been felt for motion control that balances well motion execution time, the spectral content of the driving signal, and robustness to frequency variations in the system response.